Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Real-analytic geodesics in the Mabuchi space of Kähler metrics and quantization

Alix Deleporte, Steve Zelditch

Abstract

We prove the convergence of quantized Bergman geodesics to the Mabuchi geodesics for the initial value problem, in the case of real-analytic initial data and in short time. This partially solves a conjecture of Y. Rubinstein and the last author. We also argue against the existence of a solution to the boundary value problem, generically in real-analytic regularity.

Real-analytic geodesics in the Mabuchi space of Kähler metrics and quantization

Abstract

We prove the convergence of quantized Bergman geodesics to the Mabuchi geodesics for the initial value problem, in the case of real-analytic initial data and in short time. This partially solves a conjecture of Y. Rubinstein and the last author. We also argue against the existence of a solution to the boundary value problem, generically in real-analytic regularity.
Paper Structure (15 sections, 22 theorems, 123 equations)

This paper contains 15 sections, 22 theorems, 123 equations.

Table of Contents

  1. Setting and main results
  2. The Mabuchi metric
  3. Holomorphic sections of line bundles
  4. The boundary value problem
  5. Techniques and perspectives
  6. Semiclassical analysis of Berezin-Toeplitz operators
  7. Polarisation, complexification, and complex symplectic geometry
  8. Analytic symbols
  9. Semiclassical analysis of the Bergman projector
  10. Analytic Fourier Integral Operators close to identity
  11. Approximate geodesics
  12. Geodesics as Fourier Integral Operators
  13. Geodesic equations
  14. No analytic geodesic between analytic endpoints
  15. Acknowledgements

Key Result

Proposition 1.1

Let $\phi\in \mathcal{H}$ be such that $\omega_{\phi}\in C^{\infty}$ and let $v_1,v_2\in C^{\infty}(M,\mathbb{R})$. Then

Theorems & Definitions (48)

  • Proposition 1.1: bordemann_toeplitz_1994,charles_berezin-toeplitz_2003
  • Proposition 1.2
  • Theorem A
  • Theorem B
  • Conjecture 1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 38 more